On 14 Aug 2009, at 03:21, Colin Hales wrote:
> Here's a nice pic to use in discussion.... from GEB. The map for a
> formal system (a tree). A formal system could not draw this picture.
> It is entirely and only ever 'a tree'. Humans dance in the forest.
> col
You may compare Hofstadter's picture with the Mandelbrot set, and
understand better why it is natural to think that the Mandelbrot set
(or its intersection with Q^2) to be a "creative set" in the sense of
Emil Post, that is, mainly, a (Turing) Universal system. The UD* (the
block comp multiverse) can be mapped in a similar way.
See here for a picture of the Mnadebrot set (and a comparison with
Verhulst bifurcation in the theory of chaos):
http://en.wikipedia.org/wiki/File:Verhulst-Mandelbrot-Bifurcation.jpg
Or see here for a continuos enlargement:
http://www.youtube.com/watch?v=RTuP02b_a7Y&feature=channel_page
Or perhaps better, in this context, a black and white enlargment:
http://www.youtube.com/watch?v=UrEoKFYk0Cs&feature=channel_page
or a 3-d version
http://www.youtube.com/watch?v=zciBjiD9Zfg&feature=channel_page
Colors or eights help to see the border of the set, but it is really a
subset of R^2. The border is infinitely complex, but not fuzzy! It is
really a function from R^2 to {0, 1}.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Fri Aug 14 2009 - 11:18:23 PDT
